Best Known (123, 123+24, s)-Nets in Base 3
(123, 123+24, 1641)-Net over F3 — Constructive and digital
Digital (123, 147, 1641)-net over F3, using
- 31 times duplication [i] based on digital (122, 146, 1641)-net over F3, using
- t-expansion [i] based on digital (121, 146, 1641)-net over F3, using
- net defined by OOA [i] based on linear OOA(3146, 1641, F3, 25, 25) (dual of [(1641, 25), 40879, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3146, 19693, F3, 25) (dual of [19693, 19547, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(3145, 19683, F3, 25) (dual of [19683, 19538, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3136, 19683, F3, 23) (dual of [19683, 19547, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- OOA 12-folding and stacking with additional row [i] based on linear OA(3146, 19693, F3, 25) (dual of [19693, 19547, 26]-code), using
- net defined by OOA [i] based on linear OOA(3146, 1641, F3, 25, 25) (dual of [(1641, 25), 40879, 26]-NRT-code), using
- t-expansion [i] based on digital (121, 146, 1641)-net over F3, using
(123, 123+24, 8528)-Net over F3 — Digital
Digital (123, 147, 8528)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3147, 8528, F3, 2, 24) (dual of [(8528, 2), 16909, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3147, 9851, F3, 2, 24) (dual of [(9851, 2), 19555, 25]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3146, 9851, F3, 2, 24) (dual of [(9851, 2), 19556, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3146, 19702, F3, 24) (dual of [19702, 19556, 25]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(3145, 19683, F3, 25) (dual of [19683, 19538, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(31, 19, F3, 1) (dual of [19, 18, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(3146, 19702, F3, 24) (dual of [19702, 19556, 25]-code), using
- 31 times duplication [i] based on linear OOA(3146, 9851, F3, 2, 24) (dual of [(9851, 2), 19556, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3147, 9851, F3, 2, 24) (dual of [(9851, 2), 19555, 25]-NRT-code), using
(123, 123+24, 1849541)-Net in Base 3 — Upper bound on s
There is no (123, 147, 1849542)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 13703 318820 289591 046876 180583 535650 319661 001397 805681 238312 415074 941017 > 3147 [i]